Chapter 10.5, Problem 17E

In Problems 15-18, find a formal solution to the initial boundary value problem.

$\begin{array}{l}\frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2},0<x<\pi ,0<y<\pi ,t>0\\ \frac{\partial u}{\partial x}\left(0,y,t\right)=\frac{\partial u}{\partial x}\left(\pi ,y,t\right)=0,0<y<\pi ,t>0\\ u\left(x,0,t\right)=u\left(x,\pi ,t\right)=0,0<x<\pi ,t>0\\ u\left(x,y,0\right)=f\left(x,y\right),0<x<\pi ,0<y<\pi \end{array}$

for the given function $f\left(x,y\right)$.

$f\left(x,y\right)=y$